We report on progress in measuring disconnected correlators associated with
pseudoscalar flavor-singlet mesons. This will eventually allow us to compute
the masses of the eta and eta' mesons. Flavor-singlet physics also presents an
interesting test of the staggered fermion formulation, as disconnected
correlators are sensitive to whether the same action governs both sea quarks
and valence quarks. It can also help test the validity of the ``fourth-root
trick'' used in unquenched lattice calculations where the number of flavors
$N_f<4$.Comment: Talk presented at Lattice 2005 (Hadron spectrum and quark masses), 6
  pages, 3 figure

Gregory, Eric B.

Irving, Alan C.

McNeile, Craig

Miller, Steven

Sroczynski, Zbigniew

English

arXiv

We report on progress in measuring disconnected correlators associated with pseudoscalar flavor-singlet mesons. This will eventually allow us to compute the masses of the eta and eta' mesons. Flavor-singlet physics also presents an interesting test of the staggered fermion formulation, as disconnected correlators are sensitive to whether the same action governs both sea quarks and valence quarks. It can also help test the validity of the ``fourth-root trick'' used in unquenched lattice calculations where the number of flavors $N_

Gregory, Eric B

Irving, Alan C

University of Liverpool Repository

PoS(LAT2005)083Pseudoscalar Singlet Physics with StaggeredFermionsEric B. Gregory∗, Alan C. Irving, Craig McNeile, Steven Miller, and ZbigniewSroczynskiTheoretical Physics DivisionDepartment of Mathematical SciencesUniversity of LiverpoolLiverpool, L69 7ZL, UKE-mail: egregory@liverpool.ac.ukWe report on progress in measuring disconnected correlators associated with pseudoscalar avor-singlet mesons. This will eventually allow us to compute the masses of the eta and eta’ mesons.Flavor-singlet physics also presents an interesting test of the staggered fermion formulation, asdisconnected correlators are sensitive to whether the same action governs both sea quarks andvalence quarks. It can also help test the validity of the fourth-root trick used in unquenchedlattice calculations where the number of avors N f < 4.XXIIIrd International Symposium on Lattice Field Theory25-30 July 2005Trinity College, Dublin, Ireland∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(LAT2005)083Pseudoscalar singlet physics with staggered fermions Eric B. Gregory1. IntroductionPseudoscalar flavor-singlet mesons are interesting for a number of reasons, including the rela-tion between the η ′ and pi mass difference and the U(1) axial anomaly [1, 2], and the connectionto the topological susceptibility. Unlike flavor non-singlet states, the propagator of flavor-singletmeson contains disconnected correlators, which are particularly challenging to measure preciselyin lattice simulations. Hence flavor-singlet mesons present a rich proving ground for lattice sim-ulations. To this end a number of simulations have been performed on the lattice using quenchedWilson [3], N f = 2 Wilson [4, 5, 6, 7, 8], and N f = 2 staggered fermion formulations [9, 10].Simulations utilizing modern dynamical staggered fermion gauge configurations are of partic-ular interest. The library of N f = 2 + 1 staggered fermion gauge configurations from the MILCcollaboration [12, 13] contains extremely light up and down quarks (mq/ms < 0.2), simulated atfine lattice spacings (a ≈ 0.11fm and a ≈ 0.09fm). These configurations, and other ensemblescurrently being generated provide an unprecedented opportunity to measure mixed operators nu-merically determine the masses of both the η and η ′. Additionally, the sensitivity of disconnectedcorrelators to sea quark loops offers a probe of the validity of the fourth-root trick in the staggeredfermion formulation.With this in mind, we make a preliminary presentation of measurements of pseudoscalarflavor-singlet propagators on N f = 2 + 1-flavor improved staggered fermion configurations. Asthe work is in preliminary stages, we concentrate on some theoretical and technical issues in Sec-tions 2 and 3. In Section 4 we describe our early results with some discussion of ongoing questionsand issues, which we hope our continuing work will resolve.2. Theoretical ConsiderationsThe flavor-singlet pseudoscalar meson propagator is expressed asG(x′,x) = 〈∑iqi(x′)(γ5⊗1)qi(x′)∑jq j(x)(γ5⊗1)q j(x)〉, (2.1)where the (γ5⊗1) denotes the meson as a pseudoscalar in spinor-space and a singlet in flavor-space.One can group the possible contractions into two classes: N f connected terms with contractionsconnecting fields at x and x′, and N2f disconnected terms with the qs contracted with qs at the samespace time points. So the full propagator is:G = N fC−N2f D, (2.2)where C and D are the connected and disconnected correlators, respectively. The minus sign is dueto the additional fermion loop in D. The function C is also the connected pion propagator with(γ5⊗1) operator. In full QCD, both C and G will have leading behavior that decays exponentially:C(t)∼ e−mpi t and G(t)∼ e−mη ′ t . (2.3)So the ratio of the disconnected to connected contributions to the singlet propagator behaves asR(t)≡N2f D(t)N fC(t)=N fC(t)−G(t)N fC(t)= 1−Aexp[−(mη ′−mpi)t](2.4)083 / 2PoS(LAT2005)083Pseudoscalar singlet physics with staggered fermions Eric B. Gregoryin full QCD [9]. The correct behavior of the full propagator depends on the sea quarks having thesame number of flavors and same masses as the valence quarks. If the det1/4 trick introduces someunexpected pathology into the action of the sea quarks, we might expect to detect a deviation fromthe form of equation 2.4. For example, in the limit of quenched QCD R should be linear:R(t) = A+Bt. (2.5)In non-degenerate flavor simulations one should employ obvious generalizations of equations2.2 and 2.4. For example, when N f is split into Nq +Ns flavors of light and strange quarks we have:G −→ (NqCqq +NsCss)−(N2q Dqq +N2s Dss +2NqNsDqs)R −→N2q Dqq +N2s Dss +2NqNsDqsNqCqq +NsCss. (2.6)With staggered fermions, one must rescale the disconnected correlators by 1/4 with respectto the connected correlators to take account the two loops of four staggered valence tastes in thedisconnected diagrams compared to the single valence loop in the connected diagrams[9]. In thediscussion below such rescaling is implicit in D.3. Simulation and MeasurementWe have begun measurement of singlet propagators on the following gauge ensembles:N f β L3×T am Nconfigs0 8.00 163×32 0.02 1042 7.20 163×32 0.02 2682+1 6.76 203×64 0.007, 0.05 4222+1 6.76 203×64 0.01, 0.05 644The 163× 32 configurations are small test lattices generated locally, while the 203× 64 con-figurations are part of the library of MILC “coarse lattices” [12].In principle there are two choices of flavor-singlet pseudoscalar meson operator available.These are the (γ4γ5⊗ 1) and the (γ5⊗ 1). The former is a three-link operator, with the quark andantiquark sources set on opposite corners of the spatial cube, while the latter is a four-link operator,with the quark and antiquark situated on opposite corners of the hypercube. The (γ4γ5⊗ 1) hasa parity partner, namely the scalar (1⊗ γ4γ5), which contributes an oscillating exponential to thepseudoscalar propagator. The parity partner of the (γ5⊗1), however, has exotic quantum numbersJPC = 0+−, and contributes nothing to the pseudoscalar propagator. Hence, only the (γ5⊗ 1) iswell-suited for looking at quantities such as the ratio in equation 2.4. We formulate the operators ina gauge invariant way, using symmetric covariant shifts to displace the quark and antiquark sourcesto their respective positions on the hypercube.We measure the connected diagrams using standard point sources. We measure the the discon-nected diagrams with a stochastic volume source method [14]. We define a source field η whosevalue at every lattice site is drawn from a gaussian distribution. Then we solve forO(t) =1NsrcNsrc∑i∑x,x4=t∑y,y4=tTrη†(i)y ∆γr⊗1M−1yx η(i)x , (3.1)083 / 3PoS(LAT2005)083Pseudoscalar singlet physics with staggered fermions Eric B. Gregorywhere M is the fermionic matrix and ∆γr⊗1 is the staggered meson operator that effects the four-linkshifts and the Kogut-Suskind phasing appropriate to the γr⊗1 meson. We average over a numberof different source fields η , then compute the disconnected correlator:D(∆t) = 〈O(t)O(t +∆t)〉 (3.2)In tests on the small (163×32) lattices, we found that with as few as 40 noise sources our errorwould be dominated by gauge noise even with several hundred configurations, so in subsequentruns we used Nsrc = 40. We also tested Z2 noise and found that it produced noise errors that weresystematically larger than that obtained with gaussian noise, as shown in Figure 1. Previously,for Wilson fermions Z2 noise was found to be better than Gaussian for the simplest stochasticestimators [16].0 0.02 0.04 0.06 0.08 0.11/Ns05e-081e-071.5e-072e-072.5e-073e-07error D(1)Ns=10Ns=40Ns=160Ns=640Z2 noiseGaussian noiseFigure 1: Error on the D(t = 1) for N f = 2 β = 7.2 am = 0.20 on 163× 32 lattices for gaussian and Z2noise as a function of the inverse number of sources.4. Results and DiscussionOur results to date show clear signals for disconnected and connected correlators. For the2+1-flavor ensembles, we formed the ratio:R(t) =4Dqq(t)+4Dqs(t)+1Dss2Cqq(t)+Css(t). (4.1)An example of the resulting function is plotted as the dark curve in Figure 2. Results for both2+1-flavor ensembles were qualitatively similar.It is clear that our correlators as formulated do not form the components of a singlet propagatorwhich is positive at all t, as the magnitude of the disconnected correlators exceeds that of theconnected correlators for t > 5. Correctly normalized correlators showing this behavior might083 / 4PoS(LAT2005)083Pseudoscalar singlet physics with staggered fermions Eric B. Gregory0 5 10 15t0123D(t)/C(t)  102 -   732  738 - 13621368 - 19982004 - 2628  102 - 2628 (all)Figure 2: R(t) on β = 6.76, am = 0.007,0.05 on 203× 64 lattices for full time series (black), and foursubsets (colored). Congurations indexed by trajectory number (6 trajectories per conguration).0 500 1000 1500 2000 2500trajecs-0.000500.00050.001Dqq(10)Figure 3: Time series for Dqq(t = 10) on β = 6.76, am = 0.007,0.05 on 203× 64 lattices. Same colorscheme as previous gure.illustrate a flaw in the simulated staggered sea. We have suspicions that the normalizations ofour disconnected operators may instead be incorrect, however. As we compute the disconnectedand connected correlators by different methods, we must account for different numerical factors ineach, and reconciling them is non-trivial. Further tests are being made.Further uncertainty arises from long autocorrelations in the disconnected correlators. Figure 2also shows the ratio plotted with four different subsets of the ensemble, each composed of a quarterof the time series. The large difference between bins suggests that one needs far more gaugeconfigurations to make a statement about the behavior of the D to C ratio. The autocorrelations arenot visually apparent from inspection of the time series of the disconnected correlators, e.g. Figure3, but spikes in otherwise small fluctuations seem to be enough to strongly affect D/C. It is alsopossible that with a longer time series R(t) may settle at a smaller asymptote than is apparent now.083 / 5PoS(LAT2005)083Pseudoscalar singlet physics with staggered fermions Eric B. GregoryWith these uncertainties, we do not conclude that the pseudoscalar singlet correlators producedthus far cast doubt on the staggered formulation, nor are we at this stage able to make statementsabout the lattice masses of the η and η ′. It is evident that we will first have to confirm the correctnormalizations, and then make measurements on extremely long time series ensembles. We are inthe process of generating 2+1-flavor ensembles of∼ 104 configurations on the UKQCD’s QCDOCmachine. Additionally, we are investigating different optimization strategies [9, 15] such as dilutionof the stochastic sources which may decrease disconnected correlator noise.References[1] E. Witten, Current Algebra Theorems For The U(1) ’Goldstone Boson’, Nucl. Phys. B 156 (269).[2] G. Veneziano, U(1) Without Instantons, Nucl. Phys. B 159 (213).[3] S. Itoh, Y. Iwasaki and T. Yoshie, The U(1) Problem And Topological Excitations On A Lattice, Phys.Rev. D 36, (527).[4] C. McNeile and C. Michael [UKQCD Collaboration], The eta and eta’ mesons in QCD, Phys. Lett. B491 (123) [Erratum-ibid. B 551, 391 (2003)] [arXiv:hep-lat/0006020].[5] T. Struckmann et al. [TXL Collaboration], Flavor singlet pseudoscalar masses in N(f) = 2 QCD,Phys. Rev. D 63 (074503) [arXiv:hep-lat/0010005].[6] V. I. Lesk et al. [CP-PACS Collaboration], Flavor singlet meson mass in the continuum limit intwo-flavor lattice QCD, Phys. Rev. D 67 (074503) [arXiv:hep-lat/0211040].[7] K. Schilling, H. Neff and T. Lippert, Computing the eta and eta’ mesons in lattice QCD,[arXiv:hep-lat/0401005].[8] C. R. Allton et al. [UKQCD Collaboration], Improved Wilson QCD simulations with light quarkmasses, Phys. Rev. D 70 (014501) [arXiv:hep-lat/0403007].[9] L. Venkataraman and G. Kilcup, The eta’ meson with staggered fermions,[arXiv:hep-lat/9711006].[10] J. B. Kogut, J. F. Lagae and D. K. Sinclair, Topology, fermionic zero modes and flavor singletcorrelators in finite temperature QCD, Phys. Rev. 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Pseudoscalar singlet physics with staggered fermions

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